(This topic is also in Section 1.3 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)
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Q What is a linear function?
A A linear function is one whose graph is a straight line (hence the term "linear").
Q How do we recognize a linear function algebraically?
A As follows:
A linear function is one that can be written in the form
Here is a partial table of values of the linear function f(x) = 3x - 1. Fill in the missing values and press "Check."
Plotting a few of these points gives the following graph.
The Role of b in the equation y = mx + b
Let us look more closely at the above linear function, y = 3x - 1, and its graph, shown above. This linear equation has m = 3 and b = -1.
Notice that that setting x = 0 gives y = -1, the value of b.
On the graph, the corresponding point (0, -1) is the point where the graph crosses the y-axis, and we say that b = -1 is the y-intercept of the graph
The Role of m in the equation y = mx + b
Notice from the table that the value of y increases by m = 3 for every increase of 1 in x. This is caused by the term 3x in the formula: for every increase of 1 in x we get an increase of 31 = 3 in y.
On the graph, the value of y increases by exactly 3 for every increase of 1 in x, the graph is a straight line rising by 3 for every 1 we go to the right. We say that we have a rise of 3 units for each run of 1 unit. Similarly, we have a rise of 6 for a run of 2, a rise of 9 for a run of 3, and so on. Thus we see that m = 3 is a measure of the steepness of the line; we call m the slope of the line.
Here is the graph of y = 0.5x + 2, so that b = 2 (y-intercept) and m = 0.5 (slope).
Notice that the graph cuts the y-axis at b = 2, and goes up 0.5 units for every one unit to the right. Here is a more general picture showing two "generic" lines; one with positive slope, and one wqith negative slope.
|Graph of y = mx + b|
|Positive Slope||Negative Slope|
Let y = -1.5x + 4.
Mathematicians traditionally use (delta, the Greek equivalent of the Roman letter D) to stand for "difference," or "change in." For example, we write x to stand for "the change in x."
Let us take another look at the linear equation
Now we know that y increases by 3 for every 1-unit increase in x.
Similarly, y increases by 32 = 6 for every 3-unit increase in x.
. . . .
In general, y increases by 3x units for every x-unit change in x.
|y||=||3x||Change in y = 3 Change in x|
|=||3 = slope|
Q How do these changes show up on the graph?
A Here again is the graph of y = 3x - 1 , showing two different choices for x and the associated y.
Slope of a Line
The slope of a line is given by the ratio
Definition of the Slope
For positive m, the graph rises m units for every 1-unit move to the right, and rises y = mx units for every x units moved to the right. For negative m, the graph drops |m| units for every 1-unit move to the right, and drops |m|x units for every x units moved to the right.
Fill in the slopes of the following lines.
|Getting Familiar with Slopes|
Q Two points, say (x1, y1) and (x2, y2), determine a line in the xy-plane. How do we find its slope?
A Look at the following figure.
As you can see in the figure, the rise is y = y2 - y1, the change in the y-coordinate from the first point to the second, while the run is x = x2 - x1, the change in the x-coordinate.
Computing the Slope of a Line We can compute the slope m of the line through the points (x1, y1) and (x2, y2) using
Before trying the exercises, you should go on to the next tutorial: Part B: Finding the Equation of a Line.